Tuesday, July 21, 2009

The Great Mathematical Instincts__P=NP

The Economic Mind of Charles Sanders Peirce
The Ignorance of Bourbaki, by A.R.D.Mathias
From Peirce to Skolem: A Neglected Chapter In The History of Logic, by Geraldine Brady
Lectures on Jacques Herbrand as a Logician, based on Peirce-Schroder Logics_The Godel Link
Peirce's Logic of Continuity: Existential Graphs and Non-Cantorian Continuum, by Fernando Zalamea
International Interdisciplinary Scientific and Practical Journal, REFLEXIVE PROCESSES AND CONTROL

"The mathematical laws of reasoning are, properly speaking, the laws of right reasoning only, and their actual transgression is a perpetually recurring phenomenon." George Boole
"The logician affirms, that it is impossible to deduce any conclusion from particular premises." George Boole
"... no general method for the solution of questions in the theory of probabilities can be established which does not explicitly recognise ... those universal laws of thought which are the basis of all reasoning ..." George Boole

"As a material machine economises the exertion of force, so a symbolic calculus economises the exertion of intelligence ... the more perfect the calculus, the smaller the intelligence compared to the results." William Johnson

"When I started to trace the later development of logic, the first thing I did was to look at Schröder's Vorlesungen über die Algebra der Logik, ...[whose] third volume is on the logic of relations (Algebra und Logik der Relative, 1895). The three volumes immediately became the best-known advanced logic text, and embody what any mathematician interested in the study of logic should have known, or at least have been acquainted with, in the 1890s.

While, to my knowledge, no one except Frege ever published a single paper in Frege's notation, many famous logicians adopted Peirce-Schröder notation, and famous results and systems were published in it. Löwenheim stated and proved the Löwenheim theorem (later reproved and strengthened by Thoralf Skolem, whose name became attached to it together with Löwenheim's) in Peircian notation. In fact, there is no reference in Löwenheim's paper to any logic other than Peirce's. To cite another example, Zermelo presented his axioms for set theory in Peirce-Schröder notation, and not, as one might have expected, in Russell-Whitehead notation.

One can sum up these simple facts (which anyone can quickly verify) as follows: Frege certainly discovered the quantifier first (four years before Oscar Howard Mitchell, going by publication dates, which are all we have as far as I know). But Leif Ericson probably discovered America "first" (forgive me for not counting the native Americans, who of course really discovered it "first"). If the effective discoverer, from a European point of view, is Christopher Columbus, that is because he discovered it so that it stayed discovered (by Europeans, that is), so that the discovery became known (by Europeans). Frege did "discover" the quantifier in the sense of having the rightful claim to priority; but Peirce and his students discovered it in the effective sense. The fact is that until Russell appreciated what he had done, Frege was relatively obscure, and it was Peirce who seems to have been known to the entire world logical community. How many of the people who think that "Frege invented logic" are aware of these facts?"
Hilary Putnam

"...to design logic as a calculating discipline, especially to give access to the exact handling of relative concepts, and, from then on, by emancipation from the routine claims of natural language, to withdraw any fertile soil from "cliché" in the field of philosophy as well. This should prepare the ground for a scientific universal language that looks more like a sign language than like a sound language." Ernst Schröder

"Two things here are all-important to assure oneself of and to remember. The first is that a person is not absolutely an individual. His thoughts are what he is "saying to himself," that is, is saying to that other self that is just coming into life in the flow of time. When one reasons, it is that critical self that one is trying to persuade; and all thought whatsoever is a sign, and is mostly of the nature of language. The second thing to remember is that the man's circle of society (however widely or narrowly this phrase may be understood), is a sort of loosely compacted person, in some respects of higher rank than the person of an individual organism." (5.423)2 C.S.P.

It has been duly noted that great minds exist all throughout history, but have you ever found many in agreement, except in the fields of pure and applied mathematics(not withstanding foundational exceptions of 1.number, 2.space, and 3.order.)?

If we are to ever understand a computerized computational intentionality of the spirit of humanity, we must first land in the solid ground of a sound scientific method. In order to have a sound scientific method, one must clearly identify exactly what that method is. In order for it to cover a large enough area of ideas, it must be of the nature of genericity, on the one hand, yet in order to be exactly clear, it must also be able to cover the specifically particular, on the other hand. The only language possible, to fill this technical order is simple, and or, complex mathematics, as logic herself carries far too much intellectual baggage from its long history of true, false and exaggerated interpretations, yet mathematics, or basic arithmetic, draws the ‘necessary conclusions’ we are looking for, and only she does so.

In order to use a basic arithmetic, one must specify the subject area with pure and exacting logic method, as only the basic pure logic has the tools, of course and fine grained definitions, explanations and understandings/comprehensions, required for such a potential task. Minds may understand each other in many other forms of expression, such as spirit, psychology, philosophy, sociology, etc., but only logic can cover such a vast territory as what I wish this post to express. Therefore, I’ll join the most fundamental maths and the most fundmental logics, into a fully explicated process. This process first needs a much deeper analysis and explanation, to proceed further.

Many societies, East and West, first made use of the early maths’ and logics’ systems, but our records only become fairly complete with the early Greeks, so this is where I wish to start. How did they develop their foundations of maths, logics, and knowledge method systems of? This is where it’s really quite simple__marks in the sand, and angles on their lean-to’s, plus the circles in the sky. And what did the early Greeks first work in? Lines, angles and circles. And how did they first make them? Again, marks in the sand with straight edges, strings/sticks and points(compasses), and these two created circles, from which all other accurate figures can be drawn. And all from these first figures and maths, comes the algebras of angles, areas, magnitudes and multitudes(so to speak). Now, this is of course an over-simplification, but we only need the basics of simplicity, as it’s the basis of all complexities, and therein lies our fundamental logical method__Idea, action --> method. The early Greeks, as well as all other early peoples, just looked around them, accidently found or guessed the idea(abduction) present in the real world, turned it to action(mental induction/inference/will), and created the fundamental methods(abductions/deductions/inductions) of both logics and maths.

What have we wrought on this simple perfection, except extreme excessive complexity? Nothing much of extended value, that can not be handled by these earliest of mathematical and logical ideas, methods, systems, and simple geometries. Pythagoras may not be the true father of the formula of his name, but he’s who popularized its use, so it matters not who created it, as long as we recognize its true fundamental logical property of a potential method creation. Of course, it’s not the earliest method either, but it’s the earliest most useful method, as it produced one of the world’s most useful formulas, even to this very day, and possibly created ‘pi’ on the way. Not much in the world would have been built square and aesthetically pleasing without it. All modern engineering still uses both its method example, and its simple maths and logics, spawned from it. So, the question I ask is, “Do we need an any more complex method, logic and math to truly solve for all the world’s problems, even P=NP?” And I say we do not…!__with the caveat, that they be extended through all the modern inter-relations, integrations and extensions, from their original planar geometries and logics, to spherical geometry and logic, to projective geometry and logic, and finally to matrix geometry and logic, to handle the newest complexities of the conformal logics, maths and truths. These are not as complex as has been purported, when proportionalities are recognized and realized as the basic method of all, as the original Pythagorean logic, geometry, and algebra is a fundamental universally proportional system. These are simply basic method extensions of our understanding of the fundamental Pythagorean/Archimedean/Nicomachian systems, into logarithms, linear, scalar and polar co-ordinate systems and cross-historical-time inter-relations and integrations of.

No doubt, this comes as a total shock to almost any mathematician, logician, scientist, philosopher or whomever, yet it’s what I intend to prove, in the rest of this post. It’s simply the over-complexification of methods, that’s holding the world hostage of attaining its true and full potential, and the above truths are all that’s needed to expose the fallacies of the last 5000 years of academic mis-steps, senseless arguments and her many anti-truths involved. We’ve ended, as of Sept. 08, in the overly-foolish situation where the sematic web has become completely parasitic on the pragmatic web, or in other words, where the false intellectual/academic actions have sadistically become over-parasitic and monopolistic on the true physical actions of reality. Anyone who doesn’t believe this had better review the numbers at C.H.I.P.S.(clearing house inter-bank payments system, i.e., global transactions turnovers) and the B.I.S.(bank of international settlements_i.e., derivatives transactions turnovers), then you can also look at the globe’s T.F.P.(total factor productivity) over the last year, then over the last thirty years, at the I.M.F. and World Bank. It’s rather disgusting__the last thirty+ years de-regulations into near insanity, at the least, total inanity…

Enough of the disgust, insanity and inanity. I’d like to now dedicate the rest of this post to the Peirce’s, Benjamin and Charles, as without their inter-relational algebras, logics and maths, this post would not be possible. Though Charles’ system of logic and math, founded on his father’s, DeMorgan’s, Jevons and Boole’s, was never thoroughly finished, Joseph Brent’s ‘C.S. Peirce, A Life’ offers enough information to complete it, when added to all the secondary authors’ contributions, over the last hundred years. As you may know, Charles Sanders Peirce was, and wanted to be, known as a logician, and that he certainly was. IMO, the best logician the world has ever produced, bar none. He stands far and above all the authors of our entire history, as it’s his ideas, grounded in basic instinct, and their extensions and relations that make possible a truly new computational intentionality, which can be thoroughly computerized, as a total embodied mind mathematics and logic, or as a totally new arithmetic-logic.

Though their relational systems were never fully related and transposed on historical systems, all throughout history, into the powerful system they portend, this is what I intend to show. Stated simply, C.P.’s system can unite and explain all possible mathematical and logic processes, past, present and future, in a model, having a universal standpoint of eclectic genericity, yet fully mathematize most all real particulars, at the same time. This only requires a true logical, analogical method of the math integrations, not yet seen possible.

Please download these files at: http://www.flipdrive.com/flipdrive.php to follow the rest of the text. The user name is, lgillespie , and the password is, wolf9999 , then click on My Drive, and all the files can be downloaded by clicking on them(1st 8 doc files). Sorry for the inconvenience.](Requires Microsoft Word 5+, and please view files in print preview, on MS file menu, for full and correct details) As a prelim to these graphs explanation you may want to read this: A Universal Philosophy of Pure Liberty, Natural Money, Aesthetic Ecology, and Value Validity Proofs of Peirce’s Pragmaticism and Abduction… as it’s directly related, but it’s not necessary, as I’ll give a new explanation related to this post’s graphed method direction. If interested, you may want to download the other files below the 1st eight, as they relate completely to this work. They are my comparison charts and papers of related notes, which I’m working with to write my recent posts…

Toward the end of his life, Charles worked out a new system of e-graphs and categories to compliment his life’s work in logic, to make it more easy to analogically understand his system of thought, as at the time, knowledge was not generally high enough to fully understand his complex system, which was really so simple, it seemed complex. It’s basically just his admitted deeper extensions of our basic logical and mathematical instincts, but most don’t realize how powerful our deepest basic instincts truly are, when fully explored, as C.P. did. Everyone thinks they know what fundamental logic is__they do not, right to this very day. Just to give you an idea, I’ll ask this question, “How does fundamental pure logic differentiate, or make distinctions, between its ego-self and fundamental truth?” Peirce’s method was the fundamental mathematical continuum, which he realized from Plato’s, Kant’s and Boole’s systems of the fundamental laws of mind. Now, some may say how do you find a fundamental law of mind? That’s a fair question, and one that was answered long ago by Plato, Kant and Boole. Plato’s was the esthetic nature of rationality or ratio, that is the ratios of mathematical rationality of the law systems one is living in and designing anew. This can only be had by abducting ideas from the general universal continuum of infinity, but you may ask how can one abduct or deduct a ratio from infinity? The simple trick is to start with true mathematical equilibriums, then the ratios of rights accorded to citizens is a simple proportion according to the method of designing fair laws through the rational processes of greater goods and best orders, as per Plato. This simply requires a knowledge of fairly incentivized liberty, which Plato had in abundance, as he was also a first rate economist/sociologist, for the times. It isn’t a matter of deducting from infinity, but realizing infinity can be represented just as well by arithmetic equilibriums of large magnitudes and multitudes, and simply subtracting equal proportions by ratio for the possible rights to be granted to citizens, as one for each, two for each, three for each, etc., which gives one the pure mathematics of a future state of pure liberty potentia. From there, it’s simply a matter of comparing intelectics, dialectics, intellects and politics, then adjusting the society’s laws and constitutions toward achieving a close proximity to the stated ideal incentivized liberty potentia, or simply using C.P’s ‘Metaphysics of Habits’ and ‘Science of Logics’ to design scientific methods and models of better laws, constitutions and societies.(Addendum__Remember, only math proves logic in the raw, i.e., empirically from a priori ground. In order to further understand the above infinity continuum, it’s easier to imagine a triadic infinity continuum. That would be the pure unbounded infinity, and it’s two related bounded rational infinities, i.e., the largest and smallest rational, say Cantor, numbers, or as I’m visualizing them above, as two large virtual, cardinal number series, bounded rational infinities of equal size, thus equilibriums, to draw the real number series from, just as similarly, as Kant originally may have done. He more than likely only used the space continuum of infinity to draw his numbers from to build his ‘Universal Law of Liberty’ model, but I find the explanation above more analogical, with this math model. And, in mathematics, as long as our ground is true and allows a possible real potentia, we may substitute virtual systems for real systems, just a 1 and 0 are substituted for all and none, all throughout modern maths. Even the early Greeks substituted reals for virtuals, and vice versa in their geometric algebras. And it’s only by our ability to draw(abduct) from a future infinite continuum potentia, that we are able to build new mathematical-logic model possibilities, to solve real world problems, as deducting from the past only allows trite re-arrangements of the troubled existing. All truly new ideas(hypotheses) are drawn(abducted) from the future continuum possibilia, and inducted into the present, as possible change, scientific-logic, methods and models.)

The next great mind C.P. relied on was Kant, and his similar idea, more than likely copied closely from Plato, which is translated in Kant’s ‘The Metaphysics of Law’, as ‘The Universal Law of Liberty’, which is also the same mathematical-logic method as per above. Kant describes it even better than Plato here; LINK. C.P. also relied on Boole for his, ‘An Investigation of The Laws of Thought’, which is available at LINK. Both Kant and Boole related their systems from the same continuum/space of mind which Plato used, yet Plato under the names of Forms, Archetypes and Universal Ideas and Ideals. It was simply the same ‘Internal Continuum’ he was referring to, when dealing with mathematics. All, including Peirce and most mathematicians, refer to this same continuum to draw(abduct) new ideas, ideals, methods and systems from. Formalizing methods and systems is simply the process of formulating the fairest mathematical-logical and esthetic laws and constitutions possible, as the epistemic process breakes down at this level, and the social matrix of both the internal and external continuums must be used, transposed and fused, for success, as per C.P.’s ‘Metaphysics of Habits’ and his ‘Inter-Relational Logics’. Though, this is really where C.P.’s system stopped, as he didn’t see how to use the matrix he discovered of relational systems’ logic and habits to create new law systems, and IMO, because he was continually looking at it too epistemically, when he should have realized his, ‘Metaphysics of Habits’ could be turned to creating new laws and constitutions to better society, at the ontological level of states and nations’ laws and constitutions, also. He had the complete instinctual mathematical-logic necessary, he just didn’t apply it, as he was simply concentrating far too hard on his own epistemic state of possible fame, which was the major flaw of his lifelong character, as mentioned by Brent. His ‘Metaphysics of Habits’ and ‘Inter-Relational Logics’ cross-transpose all across history, into a scientific method of arithmetic-logic, So Powerful, it can rebuild a torn and tattered world, no matter how bad you may see the future, when combined with his brilliant student T.Veblen and his studied, brilliant scholar J.M.Keynes, and on up to P.Davidson, or myself and several others, today.

‘The Missing Peirce Method’ is simply the Method of seeing his entire epistemics of self-ontology, clearly teleologically and mereologically interpreted, into a thoroughly new universal vision of ‘Geometrical Matrix Logic’, epistemics and ontology, applied to world law systems, encompassing his entire system of ‘The Metaphysics of Habits’, ‘The Science of Logic’, ‘The Economy of Research’, ‘The Economy of Philosophy’ and ‘The Economy of Ideas’. He is still the genius of genuine scientific methods and logical categorical systems’ building, when applied through our greater knowledge systems of sound ideas. Put bluntly, mind is simply extended instinct, and all we need do is understand the totality of our own instincts of interpretations of the real world, as C.S.Peirce was primarily no more than an Ideal-Realist, of the highest order. BTW, C.P. was also one of the founders of conformal projective logic and her related maths, through his quincuncial projection math, he learned through studying Grassmann and Clifford, and most likely Desargues, its truest beginnings.

Finally, let’s take a thorough look at the graphs(user name is lgillespie & password is wolf9999 then click My Drive, then click 1st 8 doc files, plus the others if you wish, as they relate) you may have downloaded. If not the rest will be too hard to follow, so please download them as per above instructions. Please use the graphs according to the numbers on the graphs, and not their file name numbers.

Graphic #1 is my model of ‘Architecture of Mind’, all the way from Plato to Peirce, on up to the present day of Dr. Paul Davidson. In this representational model is clearly shown an analogical process of mind/brain and thought actions. Perception is our knowledge generator of computational intentionality, inter-acting with all our perception agents, as per colors and boxed descriptions, plus the inter-relational arrows of essence and agent actions. To me the epistemic brain/mind is simply a triadic process through, 1.Basic emotional sensations, 2.Our 5 basic sensori organs, and 3. Our scientific, and other, intellectics, or the basic process of Peirce’s semeiotics. All is as Peirce described it in his ‘One’, ‘Two’, ‘Three’ process. This is the basic continuum of our internal epistemic agency and agents. It is an embodied mind process, and is extended into the external continuum of our personal universe, and the greater universe, yet the body/mind, IMO, is biologically limited to the speed of light, at c in vacuum. Some may disagree, that’s fine. If you care to check out the modern conformal math matrix represented, I’d suggest August Stern’s ‘Matrix Logic and Mind’. Though I don’t agree with it all, as pertains to extended mind, I do agree with most of his quantum matrix mind maths, logics and ideas.

Graph #2 actually requires graph #3 first, to offer a better understanding of the epistemic relations of #1 to #2. In #3 is shown ‘The Golden Ratio of True Liberty’, the state of law and government we are striving toward, and certainly not what we actually live in. If the Greeks had actually used their own maths’ logical truths to form their laws, constitutions and governments, the world would have long ago evolved to near scientific utopias, but the continuum has had a two-thousand+ year history of hiding these simple truths from us. This graphic is the first to show the historical cross-transposition of Peirce’s inter-relational logic abilities of explanations possible, when we fully interpret the Pythagorean’s triadic system of math, and apply it. Although the ‘Kepler Triangle’ was not part of the original G.R. drawing, it aptly fits all the way from the Pythagoreans to C.S.Peirce, J.M.Keynes and P.Davidson, thus showing the transpositional powers of vision possible about Peirce’s logic and metaphysics systems, as these graphs would not have been possible without his original sagacity. All these original Greek and Kepler idea graphics can be accessed at Wikipedia, if further interested in how they’re drawn.

In order to understand graph #1’s epistemic gap problem of free will and responsibility, just transpose/transduct back and forth between #1 and #3. When we run up against the epistemic lack of answers, we must use Peirce’s external ‘Metaphysics of Habits’ to fill the gap, as his ‘habits’ are a representation of culture’s logical laws of universal semi-liberty, from Plato and Kant, with some explanation being supplied by Boole’s logic and math continuum, also. You’ll notice the green rectangle, representing the ‘Golden Mean’ and ‘Golden Variable’. The variable is actually compliments of a friend of mine L.Stern, which actually turned out to be the best explanation of liberty’s definition, as a variable freedom within civil law and socienty. The two rectangles above and below the center one represent the space of liberty/freedom infringing on other’s rights as free citizens, of an otherwise respected liberty. Of course, it varies the two directions, at its limits and otherwise, as so shown. This understanding of the external liberty of ‘habits’ is the only truth system possible of solving the epistemic dilemma between free will and responsibility, as we should all respect a responsibility to, at the least pure liberty, if not a great deal of some parts of our present liberty system, even though full well knowing it needs changing for the better. It’s just everyone must realize, change requires the safety-net of slow sliding-time-scaled laws. Without it, we’re all dead. As I’ve drawn it, it actually represents the future state of pure liberty. For worse state representations of liberty and freedom, check graphs #7, #8 and #9. They are more or less self-explanatory, in relation to the above.

Mathematizations of the above are represented by graph #2, ‘A Triadic Cartesian Graph of Universal Logic Functions’. The mathematics of this graph was accomplished by transposing Peirce’s ‘Metaphysics of Habits’ onto Kant’s ‘Universal Law of Liberty’, through Descartes’ graphics, and can be translated by most any type of geometric algebra, i.e., planar, triganometric, polar, or conformal co-ordinates, etc. Since habits of our personal mind actually extend into the real world as our accomplishments of personal desires, ambitions and goals, which are fully mathematizable as external objects, we can enter the historical numbers necessary, and actually figure the true and entire metaphysical and logical processes of the world, and even do probability projections into the future, in general terms, as they are no more than our actions already visible in our inhabitants, homes, belongings, buildings, and businesses, but I warn, this is only a pure genericity view__as the future has many blind avenues to our possible understandings. Even all our present and past emotional actions are represented there, as emotional materialism, and fully mathematizable, whether some like it or not__It’s true. Also, all moral actions can be bumped up through esthetics(highest forms of moral actions, due to being esthetic), to rational ethics(highest forms of ethics, due to being esthetic), into fully logical actions(highest forms of rational ethics, due to being esthetic). These ideas of esthetics relations are extended representations of Peirce’s philosophical system ideas.

Graph #4 must be seen in the light of graph #3, as this is the true heart of my Peircean geometric-logic system, clearly showing the universal mathematics of not only liberty, but economics, true exchange rate mechanics, true global money and prices’ mechanics, global corporate power in relation to people’s power, the power of law, and the psychology of all involved. Sounds like a tall order, but it’s not, as you’ll clearly see just by realizing the power in inter-relational logics’ and maths’ transpostions’ potentia. Why #3 and #4 are important together is the fact that they represent the best future states possible, of all just mentioned__They are our future’s pure justice potentia. The formulas chosen are the simple ‘Pythagorean Theorem’ and Archimedes’ ‘Center of Mass’ formula. Either can be used in different areas, although the Pythagorean Theorem is the most powerful as it’s a triadic proportionality formula, thus having more representational power, and with the added benefit of being thoroughly analogical__What you see is what you get__It’s obvious truth in the eye, yet I still love Archimedes for quick calculations of many areas of systems’ logics. If you take a quick look at graphs #5 and #6, you’ll surely see the obvious comparisons. Graphs can be drawn to proportionally represent all the mathematical positions between all these extremes. For genericity’s sake, I’ve simply offered the extremes, as that’s all my method of logical genericity explanation really requires, and one must realize, I’m working mathematically only in close genericity of the facts, as some global numbers, to plug into these graphs are not available, so probable proportions must be substituted in some areas, and least squares maths, can be used for the best statistical averages of.

As to what the Pythagorean triadic squares can represent, that’s what I stated above as ‘showing the universal mathematics of not only liberty, but economics, true exchange rate mechanics, true global money and real ppp prices’ mechanics, global corporate power in relation to people’s power, the power of law in relation to corporations, people and government, and the psychologies of all involved’. And not only this, but any other triads I or others may wish to represent, such as environmental deteriation, health, and safety issues, on, on and on, etc., etc., etc. This is the potentia of the computational intentionality I mentioned at the start of this post. This simple geometric logic system is powerful enough to more than solve the world’s problems. It’s just the political hurdles the system must jump, to accomplish its true good. Yet, I think if enough people discover the clear simplicity of what’s represented here, we may have a chance of severing the stupidity of ‘Lil Ol Planet Earth’, and replacing it with true scientific methods, mathematical knowledge schools of thought, and pure sense and logic systems, for the first time in history…

I’m a true capitalist, but not the capitalism we presently have. I’m for a thoroughly changed and balanced system of laws, markets and money, capable of re-generating the yearnings of entire generations, and different classes of people, from all walks of life, of all lands on Earth. The least well-off deserve a crack at liberty just as much as the richest corporate manipulator. It’s time for the greedy, to stop starving the needy…!

The solution to all P=NP problems is solved by choosing the proper method of logics, actions and maths first, that do not exceed your computing powers. This simply means choosing simple geometric and algebraic maths and logics, to form your fundamental methods, models and ideas__then the computational math is easy enough that P=NP never becomes a problem. When complexities of too many Lie groups appear, simply convert/transpose to conformal/isomorphic proportional geometric math methods and systems, of simpler calculations. The universal law of isomorphic potentia covers all universal systems, inter-relational integrations, and formulas of…

I’ve earlier posted the complete computerized computational models of my own system’s method. It’s titled ‘Internal Exchange Clearing’, a proposal to, once and for all, rid the world of inflation, deflation, national debts, poverty, crime and war. Yes, you heard that right__LINK…

The End

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